Using LaTex and AMS-LaTex

I was kind of intimidated in the amount of time and energy I might have had to invest in learning yet another computer tool, language, text markup
language. So I held off for as long as I could being old fashioned and scribbling on paper with pen, sweat, and beer!

I was bored, out of beer, food, and money, the other night so I picked up one of my math books. I was reading an introduction (College level) to
algebra and noticed a couple things still being taught and thought, "That is so backwards!" I figured I could write some things down and maybe, just
maybe, write it up in a paper to see what sticks against the wall. Went to the arXiv and saw that everything is very well regulated. One item is the
exclusive use of LaTex.

LaTeX (... a shortening of Lamport TeX) is a document preparation system. When writing, the writer uses plain text as opposed to the formatted
text found in WYSIWYG word processors like Microsoft Word, LibreOffice Writer and Apple Pages. The writer uses markup tagging conventions to define
the general structure of a document (such as article, book, and letter), to stylise text throughout a document (such as bold and italics), and to add
citations and cross-references.

It has a little window that you enter in LaTex commands in and the output is displayed below. You can either type in free hand or, like I tend to do
here, write it off in another document then cut-n-paste.

The output is the same as one at Wikipedia for the Riemann Hypothesis. It is not really that much more difficult than doing BBcodes here at ATS. I
went on and created code for Euler's Product and an expansion of both the RH code and Euler. Bwaaahaaahaaaaa!

I hope that this is informative and if you had ever wondered what the heck it was all about that you try the Hostmath site and get some hands-on
learning of LaTex!

The Collatz Conjecture is fairly easy to explain. You probably were bored silly in school when it was presented.

Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to
obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach
1.

Seems easy, right? So try a few numbers. Restated, take any positive number, if it is even, divide it by 2 and that is your new number; if it is odd,
do the other math step; do this enough times you end up at 1.

n = 4 (yup, even), n = (4/2) = 2; (yup, even), n = (2/2) = 1

Done!

All even numbers will always ‘sink’ to one.

Why?

An even number is a number that can be evenly divided – Euclid

Modern math has a function called “modulo” which states, for even values of n, the following is true, n ≡ 0 (mod 2)

It reads: “n is congruent to zero, mod 2”

For odd numbers you can use the same notation can be used but with a different result. n ≡ 1 (mod 2)

Basically, divide any number by 2 and you will get either [0, 1] as a result.

Why bother even looking at even numbers then?!! Eventually, if the even number kicks out an odd number, you will either have already dealt with it,
or, you are going to deal with it in the future!

See what being limited to BBcodes does? If you quote the previous post you see that there are several items that converted over to what BBcode it
wants to. That is why the LaTex mark-up language was invented. So all the output will look the same. Plus a PDF can be watermarked, time-date stamped,
etc. That is what they are doing over at the arXiv. This prevents somebody else from appropriating your work. It also makes it difficult to prove that
somebody else was first even if they claim it. The problem with arXiv.org is that it is restricted (not any old shmoe can post a paper).

See, quantamagazine.com - A Long-Sought Proof, Found
and Almost Lost, for the tales of a retired statistician that proved a math problem but only knew MS Word. The proof was never review because he
never heard of LaTex! A couple mathematicians came to his rescue and posted his proof out to the arXiv and it was finally accepted!

That is what I am doing! Screwing around with Collatz, found a couple interesting things out and taking the time to learn LaTex so maybe, just maybe,
I can find a sponsor on the arXiv to see if what I found is valid or not.

This is my motivation to get old documentation here at work updated so I can shove it off to somebody else on the next step of the process and work on
my LaTex skills.

Next up, a LaTex treatment of the above mess in pretty, formatted PDF glory!

- end (this installment) -

PS - There are versions available for iPhone, Andriod, etc.

edit on 15-8-2017 by TEOTWAWKIAIFF because: proper end to a "continued"

The caption should say, "Collatz Conjecture Function" but hey, for 15 minutes of looking up various functions, I'm not going to complain! It took me
an hour to get back onto ATS to upload, way more than it took me to look up some formatting and functions.

It is so much better to see it as math. The choice is called "parity" of a number. If, when divided by two, there is no remainder, it is even
(congruent to "0" as the number divides equally). If there is a remainder of 1 then it is odd. The result of dividing by two is one or the other. Your
options are to divide by two; or, multiply by 3 and add 1 (I have a variation up as the f(n), since the result will always be even, why not divide by
two immediately? So I did).

I haven't done math in a while where I have to decipher my own scribble! This is such a vast improvement over Word's equation editor! I don't know why
I had an irrational fear of "another" language as in the end, it save so much time.

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